A Lojasiewicz Inequality in Hypocomplex Structures of $\mathbb{R}^2$

Abstract

For a real analytic complex vector field L, in an open set of R 2 , with local first integrals that are open maps, we attach a number µ ≥ 1 (obtained through Lojasiewicz inequalities) and show that the equation Lu = f has bounded solution when f ∈ L p with p > 1 + µ. We also establish a similarity principle between the bounded solutions of the equation Lu = Au + Bu (with A, B ∈ L p ) and holomorphic functions.